Prediction of visual acuity from wavefront aberrations

ABSTRACT

A method for generating a visual acuity metric, based on wavefront aberrations (WFAs), associated with a test subject and representing classes of imperfections, such as defocus, astigmatism, coma and spherical aberrations, of the subject&#39;s visual system. The metric allows choices of different image template, can predict acuity for different target probabilities, can incorporate different and possibly subject-specific neural transfer functions, can predict acuity for different subject templates, and incorporates a model of the optotype identification task.

ORIGIN OF THE INVENTION

This invention was made by one or more employees of the U.S. government.The U.S. government has the right to make, use and/or sell the inventiondescribed herein without payment of compensation, including but notlimited to payment of royalties.

FIELD OF THE INVENTION

This invention relates to visual acuity of a human or other animal,based on wavefront aberrations associated with the animal's visualimaging system.

BACKGROUND OF THE INVENTION

It is now possible to routinely measure the monochromatic aberrations ofthe human eye. However, one cannot yet measure the visual acuity thatwill result from a given set of wavefront aberrations. One reason toseek a prediction of acuity from aberrations is the possibility ofautomated objective measurement of visual acuity, and of automatedprescription of sphero-cylindrical corrections. However, it has beenshown that correcting the spherical and cylindrical components of theaberrations (equivalent to minimizing the RMS error of the wavefront)does not provide best acuity. Thus these automated procedures must awaita more sophisticated metric that can predict acuity from an arbitraryset of aberrations.

In the last decade there has been a revolution in measurement andtreatment of visual optical defects. This revolution has included thedevelopment of aberrometers simple enough to be used in the clinic,refinement of methods of laser surgery for optical correction, anddevelopment of various optical implants, notably intra-ocular lenses(IOL). In all of these, measurement and interpretation of wavefrontaberrations (WFAs) has played an important role. They are a simple andcomprehensive way of describing the state of the optical system. Inspite of this, there is at present no accepted, reliable way ofconverting WFAs to visual acuity, which is a standard measure of qualityof vision. The WFA Metric allows calculation of visual acuity fromwavefront aberrations.

What is needed is an approach, including one or more metrics, thatallows a prediction of visual acuity, for a human or other animal, basedon estimated wavefront aberrations (WFAs) measured or otherwisedetermined for the test subject. Preferably, the approach should allowacuity predictions for different optotypes, such as Sloan letters,Snellen e's, Landolt C's, Lea symbols, Chinese or Japanese charactersand others. Preferably, the approach should permit incorporation ofdifferent, possibly subject-specific, neural transfer functions.

SUMMARY OF THE INVENTION

These needs are met by the invention, which develops and applies anoptical-based and neural-based metric that allows prediction of visualacuity of the subject. For a given choice of an optotype set (e.g.,Sloan letters), an optical transfer function OTF(x,y) is generated,using Zernike polynomials and the associated Zernike coefficients and aspecification of a pupil aperture image PA(x,y) for two dimensionalcoordinates (x,y) for the subject. A generalized pupil image andassociated point spread function PSF(x,y) is computed, from which an OTFis computed.

A neural transfer function NTF(x,y) is specified, and a total transferfunction TTF(x,y) is computed as a product of the OTF and the NTF. Aproportion correct function P(k) is estimated from the neural images anda noise value, using one of three or more methods for such estimation. Aprobability criterion P(target) for measurement of visual acuity isspecified, normally between 0.5 and 0.8. A numerical procedure returns afinal index value j (final), which is converted to an estimate of acuityusing a standard logMAR calculation. The output of the logMARcomputation is a WFA metric that provides an estimate of visual acuityfor the subject.

The metric(s) developed here is designed to predict symbol acuity fromwavefront aberrations. One embodiment of the metric relies on MonteCarlo simulations of a decision process and relies on an ideal observer,limited by optics, neural filtering, and neural noise. A second metricis a deterministic calculation involving optics, symbols, and ahypothetical neural contrast sensitivity function CSF.

A WFA Metric is an algorithm for estimating the visual acuity of anindividual with a particular set of visual wavefront aberrations (WFAs).The WFAs represent arbitrary imperfections in an optical system, and caninclude low order aberrations, such as defocus and astigmatism, as wellas high order aberrations, such as coma and spherical aberration. WFAscan now be measured routinely with an instrument called an aberrometer.In modern practice, the WFAs are represented as a sum of Zernikepolynomials Z(x,y), each multiplied by a Zernike coefficient. A typicalmeasurement on the eye of a subject will consist of a list of about 16numbers, which are the coefficients of the polynomials. The WFA Metricconverts the list of numbers into an estimate of the visual acuity ofthe subject. If changes are planned to the WFA of the subject (throughsurgery or optical aids) the predicted change in visual acuity can becalculated.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an embodiment of a structure for computation of a WFAmetric according to the invention.

FIG. 2 illustrates the ten Sloan letters, expressed in a sans seriffont, that provide one of the optotype sets that can be used with theinvention.

FIGS. 3A-3E illustrate steps in creation of the OTF.

FIG. 4 graphically illustrates a representative radial neural transferfunction.

FIG. 5 is an embodiment of a procedure for evaluation of a probabilitycorrect index for one size optotype.

DESCRIPTION OF THE INVENTION 0. Notation and Terminology

In this presentation, an “image” refers to a finite discrete digitalimage represented by a two-dimensional array of integers or realnumbers. It has a width and height measured in pixels. Where the size isspecified it will be given as a list {rows,columns}. The image has aresolution measured in pixels/degree. The pixel indices of the image arex (columns) and y (rows). Images will usually be an even number ofpixels wide and tall. If the image size is {2h,2h}, then the indices xand y each follow the sequence {−h, . . . , 0, . . . , −h−1}. Thisplaces the origin of the image at the center. An image may be writtenwith explicit row and column arguments A(x,y), or without thecoordinates as A.

In this presentation, a dft refers to a two-dimensional finite discretedigital array of complex numbers representing a Discrete FourierTransform (DFT). It has a width and height measured in pixels. Where thesize is specified it will be given as a list {rows,columns}. A dft has aresolution measured in pixels/cycle/degree. The pixel indices of the dftare a (columns) and v (rows). Dfts will usually be an even number ofpixels wide and tall. If the dft size is {2h,2h}, then the indices u andv each follow the sequence {0, . . . , h−1, −h . . . , -−1}. This placesthe origin of the dft at the first pixel. This is the conventionalordering of indices in the output of the Fast Fourier Transform (FFT)operator. The FFT is a particular algorithm for implementation of theDFT. In the body of this document we refer to the DFT, but this willusually be implemented by the FFT.

In this presentation, vectors will be written with one subscript A_(k),and matrices will be written with two subscripts A_(j,k), where thefirst subscript indicates the matrix row. Frequently, we will deal withvectors or matrices whose elements are images, in which case the imagecoordinates x,y are omitted.

We make use of the notation A:B to indicate Frobenius inner product oftwo matrices

${A:B} = {\sum\limits_{y}{\sum\limits_{x}{{A\left( {x,y} \right)}{B\left( {x,y} \right)}}}}$This is useful to describe a sum over pixels of the product of twoimages. The modulus or norm of an image is given by∥A∥=√{square root over (A:A)}

1. Inputs and Output

The WFA metric has four inputs. A first input is a set of wavefrontaberrations, represented as a weighted sum of Zernike coefficientsz_(n)(x,y). A second input is a set of optotypes, represented in astandard graphic format, such as a font description, a set of rasterimages, or graphic language descriptors. One example set of optotypes isthe Sloan font for the letters {C, D, H, K, N, O, R, S, V, Z}, a setoften used in the measurement of acuity). A third input is a set oftemplates, equal in number to the number of optotypes in the set. Bydefault, the templates are derived from the optotype set and are not adistinct input. A fourth input is a set of parameters, some of which mayhave default values that are permanently stored within the program. Someparameters may be changed on every calculation of the metric, whileothers are unlikely to be changed often. The parameters are describedthroughout this description.

A single output, the visual acuity, is expressed as a decimal acuity orlog of decimal acuity (logMAR). An overall system structure is shown inFIG. 1.

2. Overview of the Algorithm

-   -   a. Generate the Optical Transfer Function (OTF)    -   b. Generate the Neural Transfer Function (NTF)    -   c. Generate the Total Transfer function (TTF)    -   d. Define the Proportion Correct function P(size)    -   e. Find the size for which P(size)≈P_(target)        Each of these steps is described in detail in the following.

3. Select a Set of Optotypes

The optotypes are a set of graphic symbols that the human observer isasked to identify in the course of an acuity test. Examples are Sloanletters, Snellen e's, Landolt Cs, Lea symbols, Chinese or Japanesecharacters, or other pictograms of various sorts. Each optotype set willhave a fixed number K of elements, and a defined size specification.

By way of example, the optotype set used here is the Sloan letters {C,D, H, K, N, O, R, S, V, Z}, with K=10. These letters are shown in FIG.2. Each Sloan letter has a stroke width MAR, expressed in minutes of arcof visual angle, and each letter is 5 MAR tall by 5 MAR wide. The sizespecification used here is Log₁₀ MAR, expressed aslog MAR(mar)=log₁₀(mar)

4. Determine the Usable Range of Optotype Sizes

The usable range will be limited by the resolution and size of the PSFimage. As discussed below, these are determined by the pupil size, thewavelength (λ), and the pupil magnification (m). If the PSF image has awidth of r, expressed in pixels, and d in degrees, the smalleststroke-width possible is one pixel, or

${\log\;{MAR}_{\min}} = {\log_{10}\left( \frac{60d}{r} \right)}$The largest stroke-width will be one fifth width of the largestcharacter, which will be one half the width of the PSF image; a marginis required to accommodate blur and to avoid wrap-around so thatlog MAR _(max)=log₁₀(6d)It is sometimes convenient to adopt a positive integer index thatcorresponds to size. One example is computing logMAR in steps of 1/20.In that scheme, the minimum and maximum indices would beindex_(min)=Ceiling(20 log MAR _(min))index_(max)=Floor(20 log MAR _(max))The index l then extends from 1 to l_(max)=index_(max)−index_(min)+1,and log MAR is given by

${{\log\;{MAR}} = \frac{l + {index}_{{mi}n} - 1}{20}},\mspace{14mu}{{{where}\mspace{14mu} l} = 1},\ldots\mspace{11mu},l_{\max}$Using the default parameters, the PSF image will have a width of 256pixels, and a width of 0.815525 deg. With these valuesindex_(min)=−14index_(max)=13The size index l will have values between 1 and l_(max)=28 for thisexample.

5. Generate the Optical Transfer Function (OTF)

The mathematical operations required to generate an optical transferfunction (OTF) from a set of Zernike polynomials are well known. Graphsof the results at several stages are shown in FIGS. 3A-3E.

-   -   a. Create the Pupil Aperture Image PA(x,y). The image is of size        {2h,2h}, where h is the half width of the pupil aperture image,        expressed in pixels,

$\begin{matrix}{{{{PA}\left( {x,y} \right)} = {{1{\;\;}{if}\mspace{11mu}{\sqrt{x^{2} + y^{2}}/h}} \leq 1}},\mspace{11mu}{{x\mspace{14mu}{and}\mspace{14mu} y\mspace{14mu}{integers}}\mspace{14mu} \in \left\{ {{- h},{\ldots\mspace{11mu} 0},{{\ldots\mspace{11mu} h} - 1}} \right\}}} \\{0\mspace{14mu}{otherwise}}\end{matrix}$

-   -   b. From the set of Zernike coefficients C={c₀, c₁, c₂, . . . ,        c_(N)} (expressed in microns), create a discrete digital image        of the Wavefront Aberration Image WA(x,y), with image size        {2h,2h}. If the Zernike polynomials z_(n)(x,y) are identified by        single index (the mode) n=0, . . . N; and if the c_(k) are the        coefficients of the individual polynomials, then

${{WA}\left( {x,y} \right)} = {\sum\limits_{n = 1}^{N}{c_{n}{z_{n}\left( {x,y} \right)}}}$We make use of the standard form of the Zernike polynomials as definedby Thibos, 2002, Jour. Of Optical Society of America.

-   -   c. Compute the Generalized pupil image GP(x,y)

${{GP}\left( {x,y} \right)} = {{{PA}\left( {x,y} \right)}{{epx}\left\lbrack {\frac{{\mathbb{i}}\; 2\;\pi}{{\lambda 10}^{- 3}}{{WA}\left( {{xv},} \right)}} \right\rbrack}}$where λ is the wavelength of light in nm used to illuminate the optotypeset.

-   -   d. Pad the image on the left and top with zeros to create an        image of size {2hm, 2hm}. The parameter m is the pupil        magnification.    -   e. Compute the Point Spread Function PSF(x,y)        PSF(x,y)=|DFT[GP(x,y)]|²        where DFT is the Discrete Fourier Transform operator.    -   f. Normalize the PSF.

${\overset{\_}{PSF}\left( {x,y} \right)} = \frac{{PSF}\left( {x,y} \right)}{{{PSF}\left( {x,y} \right)}}$

-   -   g. Compute the Optical Transfer Function OTF(u,v)        OTF(u,v)=2hmDFT[ PSF(x,y)]        This result is a complex image of size {2hm, 2hm}.        The height and width of the PSF image in degrees of visual angle        is given by

$d = \frac{h\; 360{\lambda 10}^{- 6}}{p\;\pi}$where p is the pupil diameter in mm. The height and width of the PSFimage in pixels is given byr=2hmwhere h is a half-width. The resolution of the PSF image inpixels/degree is

$v = {\frac{r}{d} = \frac{2\;\pi\; m\; p}{360\;\lambda\; 10^{- 6}}}$

6. Generate the Neural Transfer Function (NTF)

-   -   a. The Radial Neural Transfer Function RNTF(u,v) is a        two-dimensional real dft given by

${{RNTF}\left( {u,v} \right)} = {{gain}\left( {{\exp\left\lbrack {- \left( \frac{f}{f_{0}} \right)^{b}} \right\rbrack} - {{loss}\;{\exp\left\lbrack {- \left( \frac{f}{f_{1}} \right)^{2}} \right\rbrack}}} \right)}$where gain, f₀, f₁, b, and loss are parameters. An example of thisfunction is shown graphically in FIG. 4.

-   -   b, The Oblique Effect Filter OEF(u,v) is a two-dimensional real        dft given by

$\begin{matrix}{{{OEF}\left( {u,v} \right)} = {{{OEF}\left( {f,\theta} \right)} = {{1 - {\left( {1 - {\exp\left( {- \frac{f - {corner}}{slope}} \right)}} \right){\sin\left( {2\;\theta} \right)}{\;\mspace{11mu}}{if}\mspace{11mu} f}} \geq {corner}}}} \\{= {1\mspace{14mu}{otherwise}}}\end{matrix}$f=√{square root over (u ² +v ²)}θ=arctan(u,v)

where corner and slope are parameters.

-   -   c. Compute the Neural Transfer Function NTF(u,v), a        two-dimensional real dft given by        NTF(u,v)=RNTF(u,v)OEF(u,v)

7. Generate the Total Transfer Function (TTF)

The Total Transfer Function is given byTTF(u,v)=OTF(u,v)NTF(u,v)

8. Define the Proportion Correct Function P(k)

The steps in evaluation of the P(k) function are as follows, and arediagrammed in FIG. 5.

-   -   a. Given a size index 1, create K optotype images O_(k)(x,y).        This may be done by rendering images from a graphic description,        or the images may be pre-computed. Each image is of size {r,r}.        See above for a definition of the optotype size index 1.    -   b. Create the K Neural Images S_(k)(x,y) by computing the DFT of        the each optotype image O_(k), multiplying by the TTF, and        taking the inverse DFT,        S_(k)=IDFT[DFT[O_(k)]TTF]

where DFT is the DFT operation and IDFT is the Inverse DFT operation.

-   -   c. Create the K template images T_(k). By default, these are        identical to the Neural Images S_(k).    -   d. Compute the normalized templates. Each template is divided by        its norm, equal to the square root of the sum of the squares of        all its pixels.

${\overset{\_}{T}}_{k} = \frac{T_{k}}{T_{k}}$

-   -   e. Compute the matrix of normalized template cross-correlations        W_(j,k)        W_(j,k)= T _(j): T _(k)    -   f. Create an array of cross-correlations between each neural        images and each template. Note that the row indexes the neural        image and the column, the template.        R_(j,k)=S_(j): T _(k)    -   g. At this point two or more methods are available, which we        identify as methods 1 and 2.

Method 1.

-   -   i. Subtract each value from the main diagonal entry in the same        row, and divide by a factor that includes the parameter σ        (default value≈1). There are two possible versions of a matrix        D, identified by subscripts 1 and 2.

$D_{1,j,k} = \frac{R_{j,j} - R_{j,k}}{\sigma\sqrt{1 - W_{j,k}^{2}}}$

-   -   ii. The probability correct for optotype j is given by

$P_{1,j} = {\int_{- \infty}^{\infty}{{f(t)}{\prod\limits_{k \neq j}\;{{F\left( {t - D_{1,j,k}} \right)}\ {\mathbb{d}t}}}}}$where f(t) and F(t) are probability density function and cumulativeprobability

Method 2

$D_{2,j,k} = \frac{R_{j,j} - R_{j,k}}{\sigma\sqrt{2}\sqrt{1 - W_{j,k}}}$$P_{2,j} = {\prod\limits_{k \neq j}\;{F\left( D_{2,j,k} \right)}}$The final value of P is given by

$P = {\frac{1}{K}{\sum\limits_{k = 1}^{K}\; P_{k}}}$

9. Find the Size for which P≈P_(target)

The parameter P_(target) is the criterion probability for measurement ofvisual acuity. It is usually set to a value between 0.5 and 0.8. Thisvalue will depend upon the number K of optotypes and must be greaterthan 1/K (the probability of getting the right answer by guessing). Forthe Sloan letters, a default value P_(target)=0.55 is used. Variousefficient iterative procedures may be used to locate the value of sizefor which P≈P_(target). Here we describe the method of bisection, thoughother methods may be used.l_(low)=1l_(high)=l_(max)P _(low) =P(l _(low))P _(high) =P(l _(high))begin loop

${{{{If}\mspace{14mu} l_{high}} - l_{low}} = 1},\mspace{14mu}{{{exit}\mspace{14mu}{and}\mspace{14mu}{return}\mspace{14mu} l_{final}} = {l_{low} + {\frac{l_{high} - l_{low}}{p_{high} - p_{low}}\left( {p_{t} - p_{low}} \right)}}}$$l_{mid} = {{Round}\;\left\lbrack \frac{l_{high} + l_{low}}{2} \right\rbrack}$P _(mid) =P(l _(mid))If P_(mid)<P_(target),l_(low)=l_(mid)P _(low)=(l _(low))otherwisel_(high)=l_(mid)P _(high) =P(l _(high))

Go to begin loop

The returned value of l_(final) can then be converted to an acuity inlogMAR using the Equation above. This is the output of the WFA Metric.

FIG. 5 illustrates a sequence of steps of a procedure for practicing theinvention. In step 51, an OTF is generated. In step 52, an NTF isgenerated and is multiplied by the OTF, to form a TTF (step 53). In step54, a set of optotypes is and is subjected to a DFT process, in step 55.In step 56, the processed optotypes are used to form images S_(j) of theoptotypes. In step 57, the images S_(j) are used to create a set oftemplates T_(k), and normalized templates T_(k)* are created in step 58.Cross-correlations R_(j,k) of the images S_(j) and the normalizedtemplates T_(k)* are formed, in step 59. In step 60, cross-correlationsW_(k) of the normalized templates T_(k)*. Normalized difference matricesD_(j,k) are formed from the cross-correlation matrix R_(j,k) are formedin step 61, using information from the cross-correlations W_(k). and astatistical parameter σ, in step 62. In step 63, a probability Passociated with measurement of visual acuity is computed.

10. Unique Features of the WFA Metric

The WFA metric is the only known metric to compute acuity fromwavefronts that:

-   -   (i) incorporates a model of the optotype identification task    -   (ii) can predict acuity for different target probabilities    -   (iii) can predict acuity for different optotypes    -   (iv) allows user specification of optotypes    -   (v) can incorporate different and possibly subject-specific        neural transfer functions    -   (vi) can predict acuity for different subject templates        The template matching algorithm that is fundamental to this        metric may have other uses in predicting performance in        identification tasks.

Default Parameter value Unit Definition K number of optotypes k index ofoptotpye, 1, . . . , K └ 556 nm wavelength p 5 mm diameter of pupil h 64pixels half width of pupil image m 2 magnification d derived degreessize of the PSF image r 2 m h pixels size of the PSF image v r/dpixels/deg resolution of PSF image corner 13.5715 cycles/deg obliqueeffect parameter slope 3.481 oblique effect parameter gain 3.149614 NTFparameter loss 0.9260249 NTF parameter f₀ 35.869213 NTF parameter f₁5.412887 NTF parameter b 1.064181 NTF parameter l optotype size index WAimage wavefront aberration PSF image point spread function PA imagepupil aperture GP complex image generalized pupil TTF dft total transferfunction NTF dft neural transfer function OTF dft optical transferfunction OEF dft oblique effect transfer function O_(k) optotype withindex k S_(k) neural image of optotype T_(k) template with index k T_(k) normalized template with index k W_(j,k) cross-correlation betweennormalized templates R_(j,k) cross-correlation between normalizedtemplates and neural images D_(j,k) template response distribution meansP_(j) probability probability correct for optotype with index k Pprobability probability correct for optotypes of one size σ noisestandard deviation P_(target) 0.55 criterion proportion correct marminutes optotype stroke size x,y image pixel coordinates u,v dft pixelcoordinates j,k row, column indices of matrices C_(n) coefficient ofZernike polynomial n Z_(n) Zernike polynomial n

What is claimed is:
 1. A method of predicting visual acuity of animaging system, the method comprising: providing a set of K optotypes,numbered k=1,. . . , K (K≧2) of a specified size, to be used toestablish a reference set of symbols to estimate visual acuity, andproviding a description or image T_(k)(x,y) of each optotype, dependentupon location coordinates (x,y) associated with the optptype imageproduced by the imaging system; constructing a Wavefront AberrationImage WAI(x,y) that manifests selected image aberrations associated withthe imaging system; computing a generalized pupil image GPI(x,y), withnon-zero values confined to within a pupil aperture associated with theimaging system, that manifests the selected image aberrations and thatis dependent upon at least one wavelength λ of light with which theimage is viewed; computing a Point Spread Function PSF(x,y) that is anabsolute value squared of a Discrete Fourier Transform (DFT) of theimage GPI(x,y); computing a Normalized Point Spread Function NPSF(x,y),proportional to the Point Spread Function PSF (x,y), whose norm is 1:computing an Optical Transfer Function OTF(u,v), expressed as a functionof spatial frequency indice (u,v) in a Fourier transform plane, that isa Discrete Fourier Transform of the Normalized Point Spread FunctionNPSF (x,y), multiplied by a value h of original image size andmultiplied by an image magnification index m; generating a Radial NeuralTransfer Function RNTF(u,v), where RNTF(u,v) satisfies the followingconditions: (i) RNTF(u,v) is a continuous, non-negative function of aspatial frequency variable f=[u²+v²]^(1/2); (ii) RNTF(u,v) has at leastone value, f=f(max), for which NRTF(u,v) is a maximum; (iii) for0<f<f(max), RNTF(u,v) is monotonically increasing in f; (iv) forf>f(max), and RNTF(u,v) is monotonically decreasing toward 0; computinga Neural Transfer Function NTF(u,v) as a product of the Radial NeuralTransfer Function RNTF (u,v) and an Oblique Effect Filter functionOER(u,v) that compensates for viewing angle of the-original optotypeimage; generating a Total Transfer Function TTF(u,v), defined as aproduct of NTF(u,v) and the Optical Transfer Function OTF(u,v);generating a Proportion Correct index PC(j), which presents aprobability associated with a correct optotype that would be identififedby a subject having the Wavefront Aberration Image WAI(x,y) for at leastone optotype(j).
 2. The method of claim 1, wherein said step ofconstructing said Waveform Aberration Image WAI(x,y) comprises:providing a Pupil Aperture Imaging characteristic function PA(x,y) thathas a first value substantially equal to 1 within a pupil aperture of atest subject and has a second value substantially equal to 0 outside thepupil aperture of the test subject, where (x,y) are locationcoordinates; providing a set of Zernike polynomials Z_(n)(x,y) andassociated coefficients c_(n), and creating a discrete digital image aWavefront Aberration Image WAI, defined as an error sumWAI(x,y)=Σ_(n) c _(n) Z _(n)(x,y); where the coefficients c_(n) arechosen to minimize a computed error between an image provided by saidimaging system and the error sum; computing a generalized pupil imageGP, defined asGP(x,y)=PA(x,y)exp{2iWAI(x,y)/(λ/1000)}, where λ refers to a wavelengthused to illuminate the optotypes; and padding said image of each of saidoptotype on at least two adjacent edges of said image with zeroes tocreate an image of size (Δx,Δy)=(mh,mh), where h represents an originalimage size, measured in a selected direction, and m is a multiplierindex.
 3. The method of claim 1, wherein said Oblique Effect FilterOEF(u,v) is determined by a procedure comprising formingOEF(u, v) = OEF 1(f, θ) = 1 − {1 − exp {((corner) − f)/(slope)}}sin  2 θ         if  f ≥ (corner)        = 1  otherwisef=(u²+v²)^(1/2)θ=arctan(u,v), where corner and slope are parameter.
 4. The method ofclaim 1, wherein said process of generating said Proportion Correctindex PC(j) comprises: computing a neural image S_(j)(x,y) (j=1, . . . ,K) as an inverse DFT of a product of said Total Transfer FunctionTTF(u,v) multiplied by a DFT of an optotype image O_(j) (x,y); computinga cross-correlation matrix between each of the neural images S_(j) andeach of said templates T_(k), defined asR_(j,k)=S_(j)(x,y):T_(k)(x,y); computing a cross-correlation matrix ofeach normalized template with each normalized template,W_(j,k)=T_(j):T_(k); computing and normalizing a difference of matrixentries, R_(j,j)−R_(j,k), between each diagonal entry R_(j,j) and eachentry R_(j,k) in a corresponding row of the matrix {R_(j,k)}, to form anormalized difference matrix D_(j,k);D_(j,k)={R_(j,j)−R_(j,k})}/{σ{1−W_(j,k) ²}^(1/2)}, where σ is astatistical value that is provided or computed; and estimating aprobability for estimation of a correct optotype by a subject bycomputing a probability value that is either (i) a product of functionsof the matrices D_(jk), or (ii) an integral, with an integrand equal toa normal statistical function, multiplied by the product of thefunctions of the matrices D_(jk).
 5. The method of claim 1, wherein saidRadial Neural Transfer Function RNTF(u,v) is defined asRNTF(u,v)=(gain){exp[−(f/f₀)^(b)−(loss) exp[−(f/f₁)²]}, wheref=(u²+v²)^(1/2)is a spatial frequency and (gain), (loss), b, f₀ and f₁are selected parameter values.
 6. The method of claim 1, wherein saidset of optotypes includes at least one of: Sloan letters; Snellen E's;Landolt C's; and Lea symbols.